Simulation study of earthquakes based on the two-dimensional Burridge-Knopoff model with long-range interactions

Inhomogeneity effects on earthquake fault events

We present a detailed analysis of the dynamical behavior of an inhomogeneous Burridge-Knopoff model, a simplified mechanical model of an earthquake. Regardless of the size of seismic faults, a soil element rarely has a continuous appearance. Instead, their surfaces have complex structures. Thus, the model we suggest keeps the full Newtonian dynamics with inertial effects of the original model, while incorporating the inhomogeneities of seismic fault surfaces in stick-slip friction force that depends on the local structure of the contact surfaces as shown in recent experiments. The numerical results of the proposed model show that the cluster size and the moment distributions of earthquake events are in agreement with the Gutenberg-Richter law without introducing any relaxation mechanism. The exponent of the power-law size distribution we obtain falls within a realistic range of value without fine tuning any parameter. On the other hand, we show that the size distribution of both localized and delocalized events obeys a power law in contrast to the homogeneous case. Thus, no crossover behavior between small and large events occurs.

Numerical evidences of almost convergence of wave speeds for the Burridge–Knopoff model

This paper deals with the numerical approximation of a stick–slip system, known in the literature as Burridge–Knopoff model, proposed as a simplified description of the mechanisms generating earthquakes. Modelling of friction is crucial and we consider here the so-called velocity-weakening form. The aim of the article is twofold. Firstly, we establish the effectiveness of the classical Predictor–Corrector strategy. To our knowledge, such approach has never been applied to the model under investigation. In the first part, we determine the reliability of the proposed strategy by comparing the results with a collection of significant computational tests, starting from the simplest configuration to the more complicated (and more realistic) ones, with the numerical outputs obtained by different algorithms. Particular emphasis is laid on the Gutenberg–Richter statistical law, a classical empirical benchmark for seismic events. The second part is inspired by the result by Muratov (Phys Rev 59:3847–3857, 1999) providing evidence for the existence of traveling solutions for a corresponding continuum version of the Burridge–Knopoff model. In this direction, we aim to find some appropriate estimate for the crucial object describing the wave, namely its propagation speed. To this aim, motivated by LeVeque and Yee (J Comput Phys 86:187–210, 1990) (a paper dealing with the different topic of conservation laws), we apply a space-averaged quantity (which depends on time) for determining asymptotically an explicit numerical estimate for the velocity, which we decide to name LeVeque–Yee formula after the authors’ name of the original paper. As expected, for the Burridge–Knopoff, due to its inherent discontinuity of the process, it is not possible to attach to a single seismic event any specific propagation speed. More regularity is expected by performing some temporal averaging in the spirit of the Cesàro mean. In this direction, we observe the numerical evidence of the almost convergence of the wave speeds for the Burridge–Knopoff model of earthquakes.

Nature of the high-speed rupture of the two-dimensional Burridge-Knopoff model of earthquakes

The nature of the high-speed rupture or the main shock of the Burridge-Knopoff spring-block model in two dimensions obeying the rate- and state-dependent friction law is studied by means of extensive computer simulations. It is found that the rupture propagation in larger events is highly anisotropic and irregular in shape on longer length scales, although the model is completely uniform and the emergent rupture-propagation velocity is nearly constant everywhere at the rupture front. The manner of the rupture propagation sometimes mimics the successive ruptures of neighbouring 'asperities' observed in real, large earthquakes. Large events tend to be unilateral, with its epicentre lying at the rim of its rupture zone. The epicentre site of a large event is also located next to the rim of the rupture zone of some past event. Event-size distributions are computed and discussed in comparison with those of the corresponding one-dimensional model. The magnitude distribution exhibits a power-law behaviour resembling the Gutenberg-Richter law for smaller magnitudes, which changes over to a more characteristic behaviour for larger magnitudes. For very large events, the rupture-length distribution exhibits mutually different behaviours in one dimension and in two dimensions, reflecting the difference in the underlying geometry.This article is part of the theme issue 'Statistical physics of fracture and earthquakes'.

Statistical properties of the one-dimensional Burridge-Knopoff model of earthquakes obeying the rate- and state-dependent friction law

Statistical properties of the one-dimensional spring-block (Burridge-Knopoff) model of earthquakes obeying the rate- and state-dependent friction law are studied by extensive computer simulations. The quantities computed include the magnitude distribution, the rupture-length distribution, the main shock recurrence-time distribution, the seismic-time correlations before and after the main shock, the mean slip amount, and the mean stress drop at the main shock, etc. Events of the model can be classified into two distinct categories. One tends to be unilateral with its epicenter located at the rim of the rupture zone of the preceding event, while the other tends to be bilateral with enhanced "characteristic" features resembling the so-called "asperity." For both types of events, the distribution of the rupture length L_{r} exhibits an exponential behavior at larger sizes, ≈exp[-L_{r}/L_{0}] with a characteristic "seismic correlation length" L_{0}. The mean slip as well as the mean stress drop tends to be rupture-length independent for larger events. The continuum limit of the model is examined, where the model is found to exhibit pronounced characteristic features. In the continuum limit, the characteristic rupture length L_{0} is estimated to be ∼100 [km]. This means that, even in a hypothetical homogenous infinite fault, events cannot be indefinitely large in the exponential sense, the upper limit being of order ∼10^{3} kilometers. Implications to real seismicity are discussed.

Non-monotonic dependence of the friction coefficient on heterogeneous stiffness

The complexity of the frictional dynamics at the microscopic scale makes difficult to identify all of its controlling parameters. Indeed, experiments on sheared elastic bodies have shown that the static friction coefficient depends on loading conditions, the real area of contact along the interfaces and the confining pressure. Here we show, by means of numerical simulations of a 2D Burridge-Knopoff model with a simple local friction law, that the macroscopic friction coefficient depends non-monotonically on the bulk elasticity of the system. This occurs because elastic constants control the geometrical features of the rupture fronts during the stick-slip dynamics, leading to four different ordering regimes characterized by different orientations of the rupture fronts with respect to the external shear direction. We rationalize these results by means of an energetic balance argument.

Dynamical model of faulting in two dimensions and self-healing of large fractures

We describe a model for the simulation of extended two-dimensional in-plane dynamical ruptures and for the rapid calculation of statistical properties of repeated model-seismicity events. The discretization involves first- and second-nearest neighbors and is isotropic in both compression and shear properties. All rupture events obey a fracture criterion in the appropriate coordinate frame and numerical oscillations in slip velocity at crack tips due to discretization are minimized. The rupture velocities of fractures, in cases of homogeneous stress drop equal to the strength, are the supershear P-wave velocity in the direction of the prestress and the S-wave velocity in the perpendicular direction. We use the model to study the growth and healing of individual faults to understand the formation of propagating slip pulses. We confirm two mechanisms for the generation of isolated rupture pulses that have been proposed, namely, (1) a decrease in the dynamical friction with accelerating slip and (2) the encounter of the growing crack with extended regions of large difference between the threshold fracture stress and the prestress. We describe a third mechanism which is that of a velocity-dependent friction that operates equally on both the phases of increasing and decreasing slip velocities and has a characteristic length scale. It is a proxy for energy loss by radiation in a three-dimensional medium. In the case of an elongated rectangular model fault with an upper free surface and lower rigid boundary, pulses develop due to the influence of stress waves reflected from the rigid bottom boundary. In general, the excess of strength over stress drop controls crack fracture speeds; if it is too large, the crack stops. Under homogeneous stress conditions, isolated slip pulses are controlled by the spatial distribution of heterogeneities and by the velocity-dependent friction parametrization.

Observation of spatio-temporal structure in stick-slip motion of an adhesive gel sheet

We studied the sliding friction between an adhesive gel sheet and a glass substrate. In this system, the probability distribution of the force drop obeys a power law similar to that found in earthquakes and granular systems. We observed the motion of the slip regions at the frictional interfaces and obtained the spatial distributions of shear strain by image analysis. The frictional force evaluated by the image analysis is in good agreement with the actual force measured by a load cell. This indicates that the present method provides a powerful tool to study the spatio-temporal structure in the heterogeneous stick-slip motions in sliding friction.

Asperity characteristics of the Olami-Feder-Christensen model of earthquakes

Properties of the Olami-Feder-Christensen (OFC) model of earthquakes are studied by numerical simulations. The previous study indicated that the model exhibited "asperity"-like phenomena, i.e., the same region ruptures many times near periodically [T. Kotani, Phys. Rev. E 77, 010102(R) (2008)]. Such periodic or characteristic features apparently coexist with power-law-like critical features, e.g., the Gutenberg-Richter law observed in the size distribution. In order to clarify the origin and the nature of the asperity-like phenomena, we investigate here the properties of the OFC model with emphasis on its stress distribution. It is found that the asperity formation is accompanied by self-organization of the highly concentrated stress state. Such stress organization naturally provides the mechanism underlying our observation that a series of asperity events repeat with a common epicenter site and with a common period solely determined by the transmission parameter of the model. Asperity events tend to cluster both in time and in space.